Multi-finger Binary Search Trees Parinya Chalermsook Mayank Goswami László Kozma Kurt Mehlhorn Thatchaphol Saranurak Binary Search Tree (BST) search cost = depth + 1 search(2) cost = 4 Search sequence: Tree can be adapted using rotations, to prepare for next search. cost = pointer moves + rotations - offline cost: OPT - online algorithms: e.g. Splay tree [Sleator, Tarjan, 1983] Finger Search search(2) search(5) search(7) ... Search starts from previous location Tree is static, chosen optimally cost = amount of movement by finger optimal finger search cost: OPT1 Multiple fingers k fingers, stationed at nodes search starts from any of the fingers To serve request, must move some finger there search(4) Tree is static, chosen optimally cost = amount of movement by all fingers optimal k-finger search cost: OPTk Multiple fingers Our result matched by online matched by online BST with rotations BST with rotations [Cole et al., 2000] (with small overhead) [Iacono-Langerman, 2016] Our results 1. BSTs (with rotations) can simulate the k-finger optimum with small overhead: log(k) factor is optimal. 2. There is an online BST with cost: The proof has three ingredients: 1. Simulate k-finger strategy by BST with single root-pointer (and rotations) 2. Find online k-finger strategy 3. Learn optimal underlying tree 1. Simulate k-finger strategy by BST with optimal O(log k) overhead, refining previous approach with overhead O(k) [Demaine, Iacono, Langerman, Özkan, 2013] Idea: store nodes with fingers as a balanced subtree binary search tree finger tree finger tree finger pseudofinger tendon subtree O(k) fingers and pseudofingers, can store as e.g. AVL-tree tendons: almost sorted, can update in O(1) amortized time BST simulation of finger tree fingers and pseudofingers stored here e.g. AVL-tree tendons and subtrees stored here tendons stored as BST deque access finger --> search AVL tree ... O(log k) move finger --> update connected tendons insert/delete in deque ... O(1) update node role as (pseudo)finger; may trigger AVL re-balancing ... O(log k) 2. Find online k-finger strategy metric space servers requests k-server problem Goal: serve requests, minimize total movement Our task is the special case when the metric is a BST Relevant result: O(log6k) competitive online algorithm [Lee; Bubeck et al. 2018] 3. Learn best underlying tree Search sequence: - Idea: Process in epochs of length O(nlogn) - Maintain distribution over all ~4n trees When epoch starts: pick a tree from distribution, rotate to it (rotation cost amortized over epoch) When epoch ends: evaluate all trees, update distribution using multiplicative-weights-update (MWU) Loss of tree T = cost during epoch if we used T MWU guarantee nr. of possible max possible cost trees ~ 4n within epoch ~ n2 overall cost < total cost with best tree poly(n) additive term in hindsight Summary of main result 1. Simulate k-finger strategy by BST with single root-pointer (and rotations) O(logk) factor loss refines technique of [Demaine et al.] 2. Find online k-finger strategy O(log6k) factor loss first connection between k-server and BST? 3. Learn optimal underlying tree O(1+eps) factor, additive term MWU technique, randomized Applications 1. Better understand limitations of BST model [Sleator, Tarjan, 1983]: Splay tree cost is O(OPT) Dynamic Optimality Conjecture From our results: then it must be O(OPTk) logk (even k=1 case is hard to prove) barrier to Dynamic Optimality Applications 2. Understand which search sequences are easy Example: k-monotone sequence (shuffle of k runs) k-finger strategy: each finger serves one run (O(n) traversal of tree) total cost: O(nk) => BST cost: O(n klogk) Applications 2. Understand which search sequences are easy Example: weak unified bound (search is close to a recent search) amortized cost for search(xk): rank-diff. from penalty factor for recent search looking back too far we give a k-finger strategy (not so easy) OPEN QUESTIONS 1. De-randomize our online BST (randomness comes from picking tree in MWU strategy) Possible randomized/deterministic separation? 2. Towards dynamic optimality Show that Splay tree matches O(OPTk) f(k) Show that some online BST matches O(OPTk) logk (another log(k) factor unavoidable if online k-server is used).